Previous Installments: Part 1: Brownian Motion, Part 2: Liquid Crystals, Aside: Harvesting Flagella, Part 3: Optical Tweezers, Part 4: Stretching Flagella
This post combines some of the previous posts in that we are going to discuss attempts at making liquid crystals out of flagella. If you remember correctly, we are able to make liquid crystals out of straight rods by introducing what we call a depletion agent. See here for more. However, if we want to try to make liquid crystals out of flagella, we are no longer dealing with straight rods. The flagella are coiled like a spring. This creates a problem for the depletion interactions. On one hand, the flagella want to stick together because it costs less energy for them to bundle together due to the little balls floating around in the solution. But, on the other hand, they can’t bundle together and stay a fixed distance apart from each other without bending, and that bending costs energy as well. To see what I mean about the bending, imagine a piece of macaroni. If you want to put another macaroni next to it, you can do so pretty easily. If you look closely at the distance between the two macaronis, in the middle they are farther away from each other than at the ends. This can be a problem for the depletion interaction because in such an interaction, the favorable position for the two things next to each other is to be a constant distance from one another.
The top part of the image shows how the depletion interaction would want to pack curved objects. The spacing between neighboring objects is constant, but that means that each subsequent object is curved more and more tightly. The more tightly something is curved, the more energy it costs. The bottom half of the image shows packing of curved objects so that the curvature is the same for each, but the distance between adjacent objects varies. This varying of the distance between objects also has an energy cost.
The reason that I am talking about energy costs is because we are manipulating our system in such a way that it is at equilibrium. For us, we are defining equilibrium to be the lowest energy state that the system can be in. So, if something has too high of an energy cost, the system will not be in that state because there are lower energy alternatives. It is similar to how we can walk around on our toes all day, but we choose not to because it is easier (i.e. less energy) to walk with all of our foot rather than just our toes.
These two different kinds of packing will occur under different conditions. If we have very weak depletion interactions, then there will not be enough of a force to make the flagella bend in order to pack a constant distance from each other as in the top of the above image. But, if we have a very strong depletion interaction, there will be enough of a force to make the flagella pack a constant distance from each other. So, essentially, there are two ends of the spectrum and at some point in between they will transition from one state to the other.
That graph may be hard to read, but essentially, the blue line represents the energy it costs to pack as in the top of the first image, with constant spacing, the red line represents the energy cost of packing as in the bottom of the first image, and as you go from left to right, you increase the force of the depletion interaction. So, this graph just illustrates what I had said before. As the depletion interaction grows stronger, flagella are more likely to pack with constant spacing between them.
When flagella do pack with constant spacing between them, each flagella that gets added to the bundle has to bend more in order to fit. This means that each flagella added to the bundle has a higher and higher energy cost associated with it. Eventually, no more flagella will be able to join that bundle because the energy cost will be too high. This means that there is a limited size that these bundles can grow to. The maximum size of a bundle is determined by the strength of the depletion interaction; stronger depletion interaction, bigger bundle. By measuring the maximum bundle size, we can determine the strength of the depletion interaction.
This might look like just a few flagella bundled together, but is actually thousands. They have reached the maximum size of a bundle in that solution.
Another technique we have developed to measure the depletion interaction uses optical tweezers. We get a short flagella to stick to the middle of a long flagella with constant spacing between them. We then stretch the long flagella out using optical tweezers until the short flagella disconnects from the long flagella. Because we know the force at every point in time, we know the energy that it took to free the small flagella, and therefore the energy of the depletion interaction.
The top is a schematic of what is going on while the bottom is microscope images of an experiment carrying this out. Notice how as the long flagella is stretched, the small flagella is set free to float away.
These different kinds of flagella packing are interesting because they are perpetually at war with one another. The depletion interaction wants one thing and the bending of the flagella wants another. The technical term that physicists use for this is that the flagella are frustrated (honestly). There are several applications of this frustration, but this post is long enough as it is, so I think I will save the applications for another post.
Stay tuned for more information about things I am researching and working on. The next post will talk about gels that can change shape or characteristics by pulling on them.
